Table Of ContentThe Boundary Element Method
Toourwives:
ShaylaAliandLalithaRajakumar
andourchildren:
Aleef&TeeashaAli,andVinod&AnitaRajakumar
The Boundary Element Method
Applications in Sound andVibration
AshrafAli
EngineeringSolutionandSupport(ESAS),Bellevue,Washington
(FormerlywithAnsys,Inc.,Canonsburg,Pennsylvania)
CharlesRajakumar
Ansys,Inc.,Canonsburg,Pennsylvania
A.A.BALKEMAPUBLISHERS LISSE/ABINGDON/EXTON(PA)/TOKYO
Taylor & Francis
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Contents
Preface vii
Acknowledgements ix
Abbreviations xi
1 Introduction 1
1.1 Whytheboundaryelementmethod? 1
1.2 Typicalapplicationsoftheboundaryelementmethod 2
1.3 Emergenceoftheboundaryelementmethod 3
1.4 Historyofboundaryelementeigenformulations 6
1.5 Organizationofthebook 9
2 BoundaryElementMethodFundamentals 11
2.1 Introduction 11
2.2 Directmethod:weightedresiduals 12
2.3 Examples 19
2.4 Directmethod:Green’sintegraltheorem 21
2.5 Indirectmethod 23
2.6 Bodyforces 26
3 IsoparametricBoundaryElements 31
3.1 Introduction 31
3.2 Two-dimensionallinearboundaryelements 31
3.3 Higher-orderelementsin2-D 34
3.4 Boundaryelementsin3-D 36
3.5 Examples 42
4 Anisotropy,AxisymmetryandZoning 49
4.1 Introduction 49
4.2 Anisotropicmaterials 49
4.3 Axisymmetricproblems 51
4.4 Inhomogeneousregionsandzoning 54
5 Time-HarmonicAnalysisinAcousticsandElasticity 57
5.1 Introduction 57
5.2 Acoustics 57
5.3 Elasticity 62
6 DynamicAnalysis:AcousticsandElasticity 65
6.1 Introduction 65
6.2 Eigenvalueprobleminacoustics 71
6.3 Eigenvalueprobleminelasticity 72
6.4 Characteristicequationforeigenvalues 72
vi Contents
7 BasicsofAlgebraicEigenvalueProblemFormulation 77
7.1 Introduction 77
7.2 DevelopmentofBEalgebraiceigenvalueproblem 77
7.3 FormulationofInternalCellMethod 78
7.4 Exampleofinternalcellmethod:rectangularplatevibration 80
8 AlgebraicEigenvalueProbleminBoundaryElements 87
8.1 Introduction 87
8.2 Eigenproblemusingdualreciprocitymethodinacoustics 87
8.3 Eigenproblemusingparticularintegralmethodinelasticity 96
9 AdvancedConceptsinBoundaryElementAlgebraicEigenproblem 107
9.1 Introduction 107
9.2 Algebraiceigenvalueformulationusingfictitiousfunction
method 108
9.3 Exampleproblemsusingfictitiousfunctionmethod 111
9.4 Effectofinternalcollocationpointsoneigensolutions 112
9.5 Polynomial-basedparticularintegralmethod 116
9.6 Multiplereciprocitymethod(MRM) 121
9.7 Seriesexpansionmethods(SEM)withmatrixaugmentation 127
10 AcousticFluid–StructureInteractionProblems 129
10.1 Introduction 129
10.2 Boundaryelement–finiteelementcoupledeigenanalysisof
fluid–structuresystem 130
10.3 Acousticeigenproblemforenclosureswithdissipativeboundaries 137
10.4 Examplesofacousticeigenproblemwithsoundabsorption 141
11 SolutionMethodsofEigenvalueProblems 149
11.1 Introduction 149
11.2 Lanczos-basedsubspaceapproach 149
11.3 Lanczosrecursionmethod 150
11.4 Exampleproblems 157
11.5 Summarystatementsonthenon-symmetricLanczoseigensolver 160
11.6 Dampedsystemeigenvalueproblemsolution 161
11.7 Lanczostwo-sidedrecursionforthequadraticeigenvalueproblem 162
11.8 Summarystatementsoneigenvaluecomputationalgorithms 170
12 DiscussionandFutureResearch 171
12.1 Discussiononboundaryelementeigenvaluemethodologies 171
12.2 ComparisonofeigenanalysisusingBEMandFEM 172
12.3 Topicsnotcoveredinthebook 173
12.4 FutureresearchonBEMeigenanalysis 174
References 177
Index 187
Preface
The boundary element method is a powerful discretization tool in computational
mechanics. However, the eigenvalue analysis procedures within the boundary ele-
ment discretization process are still in a developing stage.To our knowledge, this is
the first-ever book dedicated entirely to the subject of boundary element eigenvalue
formulations. All the techniques of boundary element eigenvalue analysis currently
availableintheliteraturearereviewedandpresentedinthebook.Foreachtechnique,a
detailedtheoreticalformulationispresented,followedbynumericalillustrations.The
advantagesanddisadvantagesofeachmethodintermsofcomputationalefficiencies,
generalities,andformulationdifficultiesarealsopresented.
The book includes detailed formulations on linear and quadratic eigensolvers for
unsymmetricmatricessinceboundaryelementmatricesarenaturallyunsymmetric.The
book also sheds light on the ongoing debate on the choice of technique, the relative
meritsofeigenanalysesbasedontheboundaryelementandfiniteelementmethods,the
unresolvedissuesthatrequireimmediateattentionandthefuturedirectionofresearch
inthisarea.
The mode–frequency analyses of vibrating structures and the computation of res-
onant frequencies of acoustical cavities are now routinely performed in the industry.
Theeigenanalysisbasedontheboundaryelementmethodholdspromiseofbecoming
auser-friendlyandpopularprocedurewithpracticingengineerssimplybecausehere
itcanavoidthetediousandtime-consumingprocessofcreatinganadequatemeshfor
theirmodels.Someapplicationsinclude:
ElasticityArea:
1. Machines(automobiles,aircraft,etc.);
2. Machinepartssuchasconnectors,shafts,gears,fastenerssuchasscrews,pins,etc;
3. Otherequipmentthatissubjectedtovibrationsduringnormaloperation;
4. Structures(bridges,buildings,etc.).
AcousticsArea:
1. Acousticenclosuressuchasauditoriums, theaters, passenger-carandtraincabins,
etc.;
2. Hi-Fisoundequipmentsuchasloudspeakers;
3. Fluid-filledstructuressuchasoiltankers;
4. Noisecontrolofstructuressuchasautomobilemufflers;aircraftfuselages,rooms
housingvibratingmachines,etc.
Furthermore, eigenvalue analysis forms the basis for subsequent mode-based
dynamicanalyses, suchasmodesuperpositiontransientanalysis, spectrumanalysis,
andrandomvibrationanalysis.
viii Preface
Before the advent of practical numerical methods like the finite element method,
engineers conducted experiments on prototypes for determining natural frequencies.
Starting in the 1970s, computer programs based on the finite element method were
available.Althoughthefiniteelementmethodisaversatilecomputationaltechnique,
it requires a much longer data-preparation time than the boundary element method.
Specifically,engineersareforcedtospendasignificantamountoftimeingenerating
anadequatemeshforthemodelproblem.Despitetheintroductionofafewautomatic
mesh-generationalgorithmsinthecommercialfiniteelementprograms,engineersstill
continuetostruggleincreatingmeshesofacceptablequality.
Theboundaryelementmethod, ontheotherhand, isaboundarytechniquewhere
only the boundary of the domain is required to be meshed, thereby simplifying data
preparationeffortssignificantly.Amongotherbenefits,theoverallphysicaltimespent
byengineerstoperformtheanalysisisreducedsignificantly,andtheanalysisprocess
becomesmoreuser-friendly.
Thebookpresentstheeigenvalueanalysistechniquesthatusetheboundaryelement
method.The boundary element method does not easily lend itself to eigenvalue for-
mulations, especially algebraic eigenvalue formulations. Consequently, publications
on boundary element algebraic eigenvalue formulations did not come out until the
mid-1980s, although the boundary element method has been around since the late
1960s.However,non-algebraicboundaryelementeigenvalueanalysis,whichisnota
practicalanalysistechnique, appearedintheliteratureofthemid-1970s. Forhistori-
cal reasons, the book presents some materials related to the non-algebraic boundary
elementeigenvalueanalysistechniques.
Threegeneralpurposeboundaryelementcomputerprograms(GPBEST,BEASY,
andSYSNOISE)offerboundaryelementeigenvalueanalysiscapabilities.Thisbook
will hopefully satisfy the needs of engineers to acquire a detailed knowledge on the
subject.Thecapabilitiesofthecommercialprograms,suchasthosementioned,maybe
enhancedthroughtheimplementationofsomeofthedifferentmethodsofperforming
boundary element eigenvalue analysis presented in the book. This book should also
encouragethedevelopmentofnewandmorepowerfulcomputerprogramsonboundary
elementeigenvalueanalysis.
The book can be used in the graduate classes on “computational mechanics” and
“boundaryelementmethods”.Theresearchersinuniversitiesandindustries,practicing
engineers, mathematicians, computer scientists, physicists, chemists and chemical
engineers,andresearchersinbio-medicalfieldscanalsouseitasareference.
AshrafAli CharlesRajakumar
Seattle,Washington Pittsburgh,Pennsylvania
Acknowledgements
WeareindebtedtoAnsys,Inc.,Canonsburg,Pennsylvaniaforgettingusinterestedin
thesubjectofboundaryelementmethod.WearealsogratefultoSidneySolomonof
TheSolomonPressofNewYorkingivingusencouragementandvaluablesuggestions
inthepreparationandcompletionofthebook.
